where \(\mu =0\) , 1, 2, \(\ldots \)  , \(d\)  is a Lorentz index, \({D_\mu = \nabla _\mu - iqA_\mu }\) , \(F_{\mu \nu } = \nabla _\mu A_\nu - \nabla _\nu A_\mu \)  and \(\nabla _\mu \)  is the covariant derivative in the curved spacetime defined by the metric \(g_{\mu \nu }\) . Meanwhile, the extra piece \(\mathcal {L}_a\)  is introduced as containing all the extra non-minimal coupling terms between the fields and gravity that might be present in each particular realisation. As it has been discussed [1]}, [2]}, though not at all essential for the formation of the condensate, the presence of such terms is in general important in safeguarding the stability of the higher-dimensional theory after the introduction of couplings between the matter field, gravity and electromagnetism. More specifically, said stability would require the insertion of an extra set of properly weighted \(d+1\)  dimensional terms, a subject set beyond the scope of this paper; see Refs. [2]}, [4]} for a deeper insight into the issue for the case \(\ell = 2\) .
